effect of coma on the focusing of an apertured singular beam
TRANSCRIPT
ARTICLE IN PRESS
0143-8166/$ - se
doi:10.1016/j.op
�CorrespondE-mail addr
Optics and Lasers in Engineering 45 (2007) 488–494
Effect of coma on the focusing of an apertured singular beam
Rakesh Kumar Singh, P. Senthilkumaran, Kehar Singh�
Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India
Received 12 July 2006; accepted 4 October 2006
Available online 28 November 2006
Abstract
Effect of coma on the intensity distribution and encircled energy of a singular beam, at the focal plane of a lens, is evaluated
numerically by using Fresnel–Kirchhoff diffraction integral for two different values of topological charge. Results show lateral shift and
flattening of the dark core. It is noticed that the singular beam with double topological charge is affected more by comatic aberration in
comparison to the beam with single topological charge. We also verified our results by using optical transfer function approach.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Singular beam; Topological charge; Coma; Intensity distribution; Encircled energy
1. Introduction
An optical beam, possessing isolated points of undefinedphase (consequently zero amplitude) is called a singularbeam and these isolated points are called singular points.But having zero amplitude at the singular point is not asufficient criterion for a beam to be called singular.Accumulated phase on a closed loop surrounding thesingular point must be integral multiple of 2p. This integralmultiple is called topological charge (m). A cophasalwavefront containing phase singular points has the formof a one- or multi-start helical surface. One round-trip onthe continuous phase surface around the dislocation axis(axis passing through singular point or dislocation point)will lead to the next (or preceding) coil [1,2] with a pitch ml(l is the wavelength). Sign of m decides the polarity ofhelix. Due to helical structure of the constant phase surfaceit is also referred as optical vortex with positioning ofsingular point in the heart. Isolated dark core surroundedby annular rings in the transverse intensity pattern at thefocal plane of an aberration free lens is specific feature of asingular beam in contrast to non-singular beam wherebright core is surrounded by annular rings.
Due to helical wave front and specific feature of intensityprofile, a singular beam is used in trapping and manipula-
e front matter r 2006 Elsevier Ltd. All rights reserved.
tlaseng.2006.10.004
ing author. Tel.: +911126591324.
ess: [email protected] (K. Singh).
tion of microparticles [3,4]. Additional feature of a singularbeam over a Gaussian beam is that the trapped particle isexposed to lower intensities, and is thus less likely toexhibit optically induced damage. Simultaneous trappingof low-and high-index particles is possible by using asingular beam [5,6]. In manipulation or trapping, the lightfield surrounding the singularity exerts forces but theircharacteristics are determined by size and shape of the darkcore, and by the helical nature of the wavefronts associatedwith the singularity. Singular beam can also be used forastronomical application, where window of the dark coreof an optical vortex is used to examine a weak backgroundsignal hidden in the glare of bright coherent star [7].Optical processing by using vortex-producing angularphase plates is also possible [8]. This type of phase maskhas also been proposed for pattern recognition [9], byproducing a Bessel function correlation output withnarrow width.Efficiency of trap in the presence of aberration in the
focusing element demands a thorough study of aberration-induced changes in the intensity distributions. Wada et al.[10,11] investigated the role of astigmatism, and comaticaberration on the propagation characteristics of a Laguer-re–Gaussian beam. However, intensity distribution andencircled energy have not been studied for an aperturedsystem so far. The presence of geometric aberrations inoptical system [12] is known to degrade the performance ofthe traps. The effect of comatic aberration on the optical
ARTICLE IN PRESS
P'
Exit pupil plane
f
xp
yp
xg
�
θρ
ν
R. Kumar Singh et al. / Optics and Lasers in Engineering 45 (2007) 488–494 489
transfer function and intensity distribution in the focalplane of a lens has been widely studied [13–20] for non-singular beams. In the present paper, we have investigatedfor singular beams, the influence of coma of the focusinglens on the point spread function (PSF), and encircledenergy at the observation plane. The following chart isuseful to place our work in the right context. Our resultshave been verified with those of others by setting m ¼ 0 inpresence of coma and by setting m 6¼0 in the absence ofcoma.
Gaussian
Singularity z plane ygAberration(coma)
No aberration
m ¼ 0
Fig. 1. The coordinate system employed in the diffraction integral.Goodbody [13],Kapany andBurke [14],Barakat andHouston [16],Stamnes [18],Mahajan [19],Gupta et.al. [20]
Airy Pattern,Encircled energy[19]
m 6¼0
OUR WORK Swartzlander [7]2. Singular beam and aberrated system
Consider a geometry shown in Fig. 1. We assume acoordinate system with its origin at the center of the exitpupil, and z-axis along the optical axis. We chose mutuallyparallel planes orthogonal to z-axis for exit pupil and theobservation plane. Center of curvature of convergingspherical wave from the exit pupil will be located at theGaussian point P0. In presence of aberration, center ofcurvature of wave shifts from P0 due to difference betweenshape of actual wave front and ideal (aberration-free) wavefront. The actual wave front and Gaussian reference sphere[19], both pass through center of the exit pupil, and their z-axis lies along the optical axis. The amount and nature ofshift of center of curvature of actual wave front withrespect to point P0 depends on the phase modification inpresence of aberration at the exit pupil of the focusingsystem.
In this paper, we have considered one optical vortex withtopological charge m located at the origin of the coordinatesystem. A typical mathematical expression for the phasefunction with such a singularity centered at the origin is
arg ðx; yÞ ¼ y, (1)
where y denotes the angular coordinate in a polarcoordinate system. Optical beam carrying such singularityat the exit pupil plane is
Uðr; yÞ ¼ U0ðr; yÞexp ðimyÞ, (2)
where m is the topological charge and (r,y ) are the polarcoordinates on the exit pupil plane.
On the basis of factor U0 in Eq. (2) optical beam can beclassified [21] as r type (if U0 is function of radial
coordinate from the origin of coordinate system) or tanhtype (if U0 varies as tanh function) of singular beam. Non-singular beam is considered as one of the special caseswhere m ¼ 0. In this paper, for simplicity we haveconsidered U0 as constant [21], one in fact. Hence Eq. (2)is written at z ¼ 0 plane as
Uðr; yÞ ¼ exp ðimyÞ. (3)
Complex amplitude around the focal plane can becalculated by Fresnel–Kirchhoff method. The term geome-trical plane (or focal plane) corresponds to the plane, wheregeometrical focal point occurs. In presence of aberration infocusing system, the center of radius of curvature ofconverging beam is shifted from ideal focal point [19]. OnlySeidal comatic aberration is considered here, the phasemodification being proportional to the square of radialcoordinate multiplied by one transverse coordinate.Diffraction pattern of a singular beam at the focal plane
consists of central dark core surrounded by alternatingbright and dark rings with changing intensity. Central darkcore is unique feature of singular beam, and arises due todestructive interference during propagation of the beam.As mentioned earlier, accumulated phase jump is multipleof 2p around the singular point. Consequently, two pointslocated on opposite side of the same radial line posses pphase difference resulting in dark core. On the other handfor non-singular beam, intensity distribution at the focalplane consists of central bright ring surrounded byalternate dark and bright rings, i.e. airy pattern. Size ofthe dark core in the intensity pattern of a singular beamdepends on m. For a focusing system with radius a,complex amplitude at the focal plane is given [19] as
Uðr;f; f Þ ¼ C1
Z 1
0
Z 2p
0
UðyÞexp �ika
frr cos ðy� fÞ
� �
� r dr dy, ð4Þ
where
C1 ¼a2
ilfeikf exp ip
r2
lf
� �¼ C0exp i k f þ ip
r2
lf
� �.
ARTICLE IN PRESSR. Kumar Singh et al. / Optics and Lasers in Engineering 45 (2007) 488–494490
Here y is the angular coordinate in the polar coordinatesystem on the exit pupil plane of radius a. A point on theexit pupil is represented by position vector rp and normal-ized radial coordinate r by rp/a. In the integral (Eq. (4)) f isthe distance of observation plane (focal plane) from the exitpupil and k ¼ 2p/l, l being wavelength of light. Foraberrated optical system Eq. (4) can be modified as
Uðv; f Þ ¼ C1
Z 1
0
Z 2
0
UðyÞexp fi½kW ðr; yÞ�g
� exp ½�ipnr cosðy� fÞ�rdrdy, ð5Þ
where W(r,y) is the aberration function representing thewave aberration of the optical wave front at a point in theplane of exit pupil with normalized position coordinate r.(n,f) are polar coordinates at the focal plane, and n ¼ (2a/lz)r. The aberration coefficient is assumed to be in units ofwavelength. The wave aberration corresponding to coma isgiven by
W ¼ Acr3cos y,
where Ac is comatic aberration coefficient. Complexamplitude at the focal plane in the presence of coma is
Uðv;f; zÞ ¼ C1
Z 1
0
Z 2p
0
exp ðimyÞ exp ið2p=lÞ Acr3cos y� ��
� exp ½�ipnrcos ðy� fÞ�rdrdy. ð6Þ
The phase factor of the multiplicative term C1 does notplay any role in the intensity and can be ignored. In theintensity distribution, normalization factor a4
l2f 2
� �can be ignored and this result is used as intensity PSF.Intensity distribution (PSF) at the observation plane,therefore, is
Iðv;f; f Þ ¼ Uðv;f; f Þ 2. (7)
The intensity distribution derived by using Fresnel–Kirchh-off diffraction integral can also be derived by using optical
Fig. 2. Intensity distribution of singular beam with m ¼ 1 and coma Ac (a) 0.0 (
coma Ac (e) 0.0 (f) 0.5 (g) 1.0 (h) 1.5.
transfer function (OTF) approach. Origin of this statementarises due to functional relationship between OTF andPSF, i.e. OTF is the Fourier transform of PSF [15]. OTF isdefined as the normalized autocorrelation of the pupilfunction [17,22]. This approach can also be used forcalculating the intensity distribution and encircled energyat any observation plane. Since encircled energy is real, wehave to take only real part of the transfer function forcalculation of the encircled energy by transfer functionapproach [15]. Intensity distribution I(n,f) according totransfer function approach [20] is
Iðv;fÞ ¼ ð1=pÞZ 2
0
Z 2p
0
ðs;jÞexp ½inrcos ðj� fÞs ds dj.
(8)
Here C(s,j) is the autocorrelation of the apertureamplitude distribution in presence of aberration, s is thenormalized distance between two points in the apertureplane, f is the azimuthal angle in the pupil function afteraxis transformation and f is azimuthal angle on theobservation plane.To get more information about the effect of aberration,
encircled energy which is defined as the total energy lyingwithin a circle of specified radius at the observation plane,is also useful. We have considered encircled energy within acircular region of radius n0 centered at the geometrical focalpoint. This energy can be written [14] as
Eðv0; f Þ ¼
Z v0
0
Z 2p
0
Iðv;fÞv dv df. (9)
Functional relation between PSF and OTF is shown byflow chart. Here two-dimentional (2D) FT and 2D IFTstand for 2D Fourier and inverse Fourier transforms,respectively. Results obtained by these two approacheshelp us to cross-check the correctness of the computationprocess.
b) 0.5 (c) 1.0 (d) 1.5. Intensity distribution of singular beam with m ¼ 2 and
ARTICLE IN PRESS
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
Ac=
0.0
0.5
1.0
1.5
0.6
0.5
I
I
0.4
0.3
0.2
0.1
0-4 -3 -2 -1
ν0 2 31 4
m = 1
� = 0
1
0.9
0.8
0.7
Ac=
0.0
0.5
1.0
1.5
0.6
0.5I
0.4
0.3
0.2
0.1
0-4 -3 -2 -1
ν0 2 31 4
m = 1
� = π /4
1
0.9
0.8
0.7
Ac=
0.0
0.5
1.0
1.5
0.6
0.5
0.4
0.3
0.2
0.1
0-4 -3 -2 -1
ν0 2 31 4
m = 1
� = π /2
a
c
b
Fig. 3. (a) Intensity profile for optical beam of m ¼ 1 in the presence of coma (Ac ¼ 0, 0.5, 1.0 and 1.5), at the focal plane for azimuthal angle f ¼ 0.
(b) Intensity profile for optical beam of m ¼ 1 in the presence of coma (Ac ¼ 0, 0.5, 1.0 and 1.5), at the focal plane for azimuthal angle f ¼ p/4.(c) Intensity profile for optical beam of m ¼ 1 in the presence of coma (Ac ¼ 0, 0.5, 1.0 and 1.5), at the focal plane for azimuthal angle f ¼ p/2.
R. Kumar Singh et al. / Optics and Lasers in Engineering 45 (2007) 488–494 491
ARTICLE IN PRESS
1
0.9
0.8
0.7
Ac=0
Ac=
m=0
0.0
0.5
1.0
1.5
0.6
0.5E0
0.4
0.3
0.2
0.1
0
0 2 4 6
ν'
8 10 12
m = 1
Fig. 4. Encircled energy at the focal plane for beam with m ¼ 1 in the
presence of coma.
R. Kumar Singh et al. / Optics and Lasers in Engineering 45 (2007) 488–494492
Pupil function 2 D F.T.
2 D I .F.T.
Complex
amplitude (F)
Point spread
function FF*(PSF)
Autocorrelation
Optical transfer function
(OTF)
2 D F . T .
2 D I .F.T.
Multiplication
by complex
conjugate
3. Results and discussion
Intensity distribution at the focal plane is calculated byusing Eqs. (6) and (7). In the aberration-free case, integral(6) is transformed to mth-order Hankel transform [23,24],order of the Hankel transform corresponding to thetopological charge of the singular beam. However,analytical solution of integral (6) is not possible for anyarbitrary value of the comatic aberration coefficient Ac.Results of the intensity distribution are presented in Fig. 2,for m ¼ 1 and 2, and various values of Ac. Encircled energyat the focal plane has been calculated by using Eq. (9).
By using the functional relationship between PSF andOTF, we calculated OTF for the case of non-singular andsingular beams, for different combinations of the topolo-gical charge and comatic aberration coefficients. Intensitydistribution for different azimuthal angles, and encircledenergy by using two approaches mentioned earlier inSection 2 have been computed and there is agreementbetween the results so obtained. We may mention that form ¼ 0, our results for the case of non-singular beam in thepresence of coma agreed with those of Kapany and Burke[14], Barakat and Houston [17] and Mahajan [19]. Resultsfor the intensity profile and encircled energy for non-singular beam also agreed with the results given byStamnes [18], Mahajan [19] and Gupta et al. [20].
Results of PSF for aberration-free and aberrated casesare shown in Figs. 2(a)–(d). Due to non-rotationallysymmetric nature of comatic aberration, we have evaluatedintegral (6) for three different values of the azimuthal anglef. Results of intensity profile for singular beam of unittopological charge are shown in Figs 3(a)–(c) for azimuthalangles f ¼ 0, p/4, and p/2, respectively. Case Ac ¼ 0 inthese figures corresponds to aberration-free case and theresult agree with those reported by Swartzlander [7].
For f ¼ 0 (Fig. 3a) and Ac 6¼0, intensity minimum shiftsfrom the position where it occurs in the aberration-freecase. For azimuthal angles of p/4 (Fig. 3b) and p/2(Fig. 3c), minimum value of the intensity no longer remainszero and also decreases with increasing value of Ac. As thevalue of comatic aberration coefficient Ac increases, thedark core of the pattern is stretched along the x-axis. Forf ¼ p/2, the intensity distribution is symmetric withrespect to n ¼ 0, for all values of Ac. Encircled energy forsingular beam at the focal plane is evaluated by using
Eq. (9) and the results for m ¼ 1 and different value of Ac
are given in Fig. 4. These results are in accordance with theintensity profiles at the Gaussian plane in case of m ¼ 1(Fig. 3).Results of PSF for m ¼ 2 and Ac ¼ 0, 0.5, 1.0 and 1.5 are
shown in Figs. 2(e)–(h) respectively. Intensity distributionsfor three values of the azimuthal angles are shown inFigs. 5(a)–(c). For Ac ¼ 0, the dark core in the case ofm ¼ 2 is broader than in case of m ¼ 1, and the result is inagreement with that of Swartzlander [7]. In Fig. 5(a) forAc ¼ 0.5, the dark core is displaced from the origin and theintensity distribution becomes asymmetric. The intensitymaximum is increased on one side of the dark core anddecreased on the other side. For higher values of Ac, firstintensity maximum after the central dark core is furtherreduced on one side of the origin, leading to disappearanceof dip from the center. With increase in Ac for f ¼ p/4,intensity of the first maximum reduces continuously. Onthe other hand for f ¼ p/2, symmetry is maintained inintensity distribution around n ¼ 0. With increasing Ac, forf ¼ p/2, dip in intensity profile becomes less pronounced.The results for encircled energy for four values of Ac are
shown in Fig. 6, and are consistent with the results ofFigs. 5(a)–(c). In Figs. 4 and 6 note that pn ¼ n0. As anexample, for the case of Ac ¼ 1.0, reduction in intensity offirst ring is balanced by an increase in intensity value
ARTICLE IN PRESS
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
Ac=
0.0
0.5
1.0
1.5
0.6
0.5
I
I
0.4
0.3
0.2
0.1
0
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0-4 -3 -2 -1
ν0 2 31 4
m = 1
� = 0
Ac=
0.0
0.5
1.0
1.5
I
-4 -3 -2 -1
ν0 2 31 4
m = 2
� = π /4
1
0.9
0.8
0.7
Ac=
0.0
0.5
1.0
1.5
0.6
0.5
0.4
0.3
0.2
0.1
0-4 -3 -2 -1
ν0 2 31 4
m = 2
� = π /2
a
c
b
Fig. 5. (a) Intensity profile for optical beam of m ¼ 2 in the presence of coma (Ac ¼ 0, 0.5, 1.0 and 1.5), at the focal plane for azimuthal angle f ¼ 0.
(b) Intensity profile for optical beam of m ¼ 2 in the presence of coma (Ac ¼ 0, 0.5, 1.0 and 1.5), at the focal plane for azimuthal angle f ¼ p/4.(c) Intensity profile for optical beam of m ¼ 2 in the presence of coma (Ac ¼ 0, 0.5, 1.0 and 1.5), at the focal plane for azimuthal angle f ¼ p/2.
R. Kumar Singh et al. / Optics and Lasers in Engineering 45 (2007) 488–494 493
ARTICLE IN PRESS
1
0.9
0.8
0.7
Ac=
0.0
0.5
1.0
1.5
0.6
0.5E0
0.4
0.3
0.2
0.1
00 2 4 6
ν'
8 10 12
m = 2
Fig. 6. Encircled at the focal plane for energy beam with m ¼ 2 in the
presence of coma.
R. Kumar Singh et al. / Optics and Lasers in Engineering 45 (2007) 488–494494
between two separated dark regions, thus leading to aninitial increase in the encircled energy near the center.However, for the case of Ac ¼ 1.5 reduction in the intensityof the first bright ring is not completely compensated byincrease in intensity between the separated dark regions.Hence, increase in encircled energy is slow near the center,in comparison to aberration-free case.
4. Conclusion
Using the Fresnel–Kirchhoff diffraction approach, theintensity distribution and encircled energy of a singularbeam at the focal plane of a lens, has been numericallyevaluated in the presence of comatic aberration. Study hasbeen made for two values of topological charge. Thecomatic aberration results in lateral shift and flattening ofthe dark core of the intensity distribution. The effect of thecomatic aberration is more pronounced for higher value oftopological charge. The results have been verified by theoptical transfer function approach.
Acknowledgments
Rakesh Kumar Singh thankfully acknowledges thefinancial support from the Council of Scientific andIndustrial Research India (CSIR).
References
[1] Basistiy IV, Soskin MS, Vasnetsov MV. Optical wavefront disloca-
tions and their properties. Opt Commun 1995;119:604–12.
[2] Allen L, Barnett SM, Padgett MJ, editors. Optical angular
momentum. Bristol: Institute of Physics Publishing; 2003.
[3] Gahagan KT, Swartzlander Jr. GA. Optical vortex trapping of
particles. Opt Lett 1996;21:827–9.
[4] Gahagan KT, Swartzlander GA. Trapping of low-index microparti-
cles in an optical vortex. J Opt Soc Am B 1998;15:524–34.
[5] Gahagan KT, Swartzlander Jr. GA. Simultaneous trapping of low-
index and high-index microparticles observed with an optical-vortex
trap. J Opt Soc Am B 1999;16:533–7.
[6] Lee WM, Ahluwalia BPS, Yuan XC, Cheong WC, Dholakia K.
Optical steering of high and low index microparticles by manipulating
an off-axis optical vortex. J Opt A Pure Appl Opt 2005;7:1–6.
[7] Swartzlander Jr. GA. Peering into darkness with a vortex spatial
filter. Opt Lett 2001;26:497–9.
[8] Crabtree K, Davis JA, Moreno I. Optical processing with vortex
producing lenses. Appl Opt 2004;43:1360–7.
[9] Davis JA, Haavig LL. Cottrell DM Bessel function output from an
optical correlator. Appl Opt 1997;36:2376–9.
[10] Wada A, Ohtani T, Miyamoto Y, Takeda M. Propagation analysis of
the Lagurre–Gaussian beam with astigmatism. J Opt Soc Am A
2005;22:2746–55.
[11] Wada A, Ohminato H, Yonemura T, Miyamoto Y, Takeda M. Effect
of comatic aberration on the propagation characteristics of
Laguerre– Gaussian beams. Opt Rev 2005;12:451–5.
[12] Roichman Y, Waldron A, Gardel E, Grier DG. Optical traps with
geometric aberrations. Appl Opt 2006;45:3425–9.
[13] Goodbody AM. The influence of coma on the response function of an
optical system. Proc Phys Soc 1960;75:677–88.
[14] Kapany NS, Burke JJ. Various image assessment parameters. J Opt
Soc Am 1962;52:1351–61.
[15] Barakat R, Houston A. Reciprocity relation between the transfer
function and total illuminance. J Opt Soc Am 1963;53:1244–9.
[16] Barakat R, Houston A. Diffraction effects of coma. J Opt Soc Am
1964;54:1084–7.
[17] Barakat R, Houston A. Transfer function of an optical system in the
presence of off-axis aberrations. J Opt Soc Am 1965;55:1142–8.
[18] Stamnes JJ. Waves in focal regions. Bristol and Boston: Adam Hilger;
1986.
[19] Mahajan VN. Optical imaging and aberrations, Part 2 wave and
diffraction optics. Bellingham, WA: SPIE Press; 2001.
[20] Gupta AK, Singh RN, Singh K. Diffraction images of extended
circular targets in presence of coma. Can J Phys 1977;55:1025–32.
[21] Rozas D, Law CT, Swartzlander Jr. GA. Propagation dynamics of
optical vortices. J Opt Soc Am B 1997;14:3054–65.
[22] Wilson RG, McCreary SM, Thompson JL. Optical transformations
in three-space: simulations with a PC. Am J Phys 1992;60:46–9.
[23] Jaroszewicz Z, Kolodziejczyk A. Zone plates performing generalized
Hankel transforms and their metrological applications. Opt Commun
1993;102:391–6.
[24] Tranter CJ. Bessel functions with some physical applications.
London: The English Univ. Press Ltd.; 1968.